If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after t seconds is given by H = 10t - 1.86t2. a. Find the velocity of the rock after one second. b. Find the velocity of the rock when t = a. c. When will the rock hit the surface? d. With what velocity will the rock hit the surface?
> Determine whether f’(0) exists. |x sin if x# 0 f(x) = if x= 0
> The graph shows the influence of the temperature T on the maximum sustainable swimming speed S of Coho salmon. a. What is the meaning of the derivative S’(T)? What are its units? b. Estimate the values of S’(15) and
> The quantity of oxygen that can dissolve in water depends on the temperature of the water. (So thermal pollution influences the oxygen content of water.) The graph shows how oxygen solubility S varies as a function of the water temperature T. a. What is
> The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q = f(p). a. What is the meaning of the derivative f’(8)? What are its units? b. Is f’(8) positive or negative? Explain.
> Let H(t) be the daily cost (in dollars) to heat an office building when the outside temperature is t degrees Fahrenheit. a. What is the meaning of H’(58)? What are its units? b. Would you expect H’(58) to be positive or negative? Explain.
> The number of bacteria after t hours in a controlled laboratory experiment is n = f(t). a. What is the meaning of the derivative f’(5)? What are its units? b. Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you th
> The cost of producing x ounces of gold from a new gold mine is C = f(x) dollars. a. What is the meaning of the derivative f’(x)? What are its units? b. What does the statement f’(800) = 17 mean? c. Do you think the values of f9sxd will increase or decr
> If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as V(t) = 100,000(1 – 1/60t)2 0 ≤ t ≤ 60 Find t
> The cost (in dollars) of producing x units of a certain commodity is C(x) = 5000 + 10x + 0.05x2. a. Find the average rate of change of C with respect to x when the production level is changed i. from x = 100 to x = 105 ii. from x = 100 to x = 101 b. F
> The table shows values of the viral load V(t) in HIV patient 303, measured in RNA copies/mL, t days after ABT-538 treatment was begun. a. Find the average rate of change of V with respect to t over each time interval: i. [4, 11] ii. [8, 11] iii. [11
> Determine the infinite limit. 1 lim x>(7/2)+ x sec x
> The table shows world average daily oil consumption from 1985 to 2010 measured in thousands of barrels per day. a. Compute and interpret the average rate of change from 1990 to 2005. What are the units? b. Estimate the instantaneous rate of change in 2
> The number N of locations of a popular coffeehouse chain is given in the table. (The numbers of locations as of October 1 are given.) a. Find the average rate of growth i. from 2006 to 2008 ii. from 2008 to 2010 In each case, include the units. What
> Researchers measured the average blood alcohol concentration C(t) of eight men starting one hour after consumption of 30 mL of ethanol (corresponding to two alcoholic drinks). a. Find the average rate of change of C with respect to t over each time int
> A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F. The graph shows how the temperature of the turkey decreases and eventually approaches room
> A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater or less than the rate of change after an hour?
> A particle moves along a straight line with equation of motion s = f(t), where s is measured in meters and t in seconds. Find the velocity and the speed when t = 4. f(t) = 10 + 45/t + 1
> A particle moves along a straight line with equation of motion s = f(t), where s is measured in meters and t in seconds. Find the velocity and the speed when t = 4. f(t) = 80t - 6t2
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case. sin 0 lim 0n/6 0 – T/6 피/
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case. cos(T + h) + 1 lim h
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case. - 4 lim x1/4 X
> Determine the infinite limit. lim In(sin x) X0+
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case. х — 64 lim 22 х — 2
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case. -2+h – e-2 lim h0
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case. 9 + h – 3 lim h0 h
> Find f’(a). f(x) = 4/ 1 − x
> Find f’(a). f(x) = 1 − 2x
> Find f’(a). f(x) = x-2
> Find f’(a). f(t) = 2t + 1/t + 3
> Find f’(a). f(t) = 2t3 + t
> Find f’(a). f(x) = 3x2 - 4x + 1
> a. If G(x) = 4x2 - x3, find G’(a) and use it to find equations of the tangent lines to the curve y = 4x2 - x3 at the points (2, 8) and (3, 9). b. Illustrate part (a) by graphing the curve and the tangent lines on the same screen.
> Determine the infinite limit. lim In(x? – 9)
> a. If F(x) = 5x/(1 + x2), find F’(2) and use it to find an equation of the tangent line to the curve y = 5x/(1 + x2) at the point (2, 2). b. Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> If g(x) = x4 - 2, find g’(1) and use it to find an equation of the tangent line to the curve y = x4 - 2 at the point (1, -1).
> If f(x) = 3x2 - x3, find f’(1) and use it to find an equation of the tangent line to the curve y = 3x2 - x3 at the point (1, 2).
> Sketch the graph of a function f where the domain is (-2, 2), f’(0) = -2, lim x→-2 f(x) = ∞, f is continuous at all numbers in its domain except (1, and f is odd.
> Sketch the graph of a function g that is continuous on its domain (-5, 5) and where g(0) = 1, g’(0) = 1, g’(-2) = 0, limx→-5+ g(x) = ∞, and limx→5- g(x) = 3.
> Sketch the graph of a function g for which g(0) = g(2) = g(4) = 0, g'(1) = g'(3) = 0, g'(0) = g'(4) = 1, g'(2) = –1, lim,-»- g(x) lim,-- g(x) = 00, and = -00.
> Sketch the graph of a function f for which f(0) = 0, f’(0) = 3, f’(1) = 0, and f’(2) = -1.
> If the tangent line to y = f(x) at (4, 3) passes through the point (0, 2), find f(4) and f’(4).
> If an equation of the tangent line to the curve y = f(x) at the point where a = 2 is y = 4x - 5, find f(2) and f’(2).
> Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = 23 and g’(5) = 4.
> Determine the infinite limit. x, lim →3 (х — 3)5
> For the function f graphed in Exercise 18: a. Estimate the value of f’(50). b. Is f’(10) > f’(30)? c. Is f’(60) > f(80) – f(40)/80 - 40? Explain.
> The graph of a function f is shown. a. Find the average rate of change off on the interval [20, 60]. b. Identify an interval on which the average rate of change off is 0. c. Which interval gives a larger average rate of change, [40, 60] or [40, 70]? d
> For the function g whose graph is given, arrange the following numbers in increasing order and explain your reasoning: g'(-2) g'(0) g'(2) g'(4) y. y=g(x) 1 3 4 2.
> The displacement (in feet) of a particle moving in a straight line is given by s = 1/2t2 - 6t + 23, where t is measured in seconds. a. Find the average velocity over each time interval: i. [4, 8] ii. [6, 8] iii. [8, 10] iv. [8, 12] b. Find the ins
> The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s = 1/t2, where t is measured in seconds. Find the velocity of the particle at times t = a, t = 1, t = 2, and t = 3.
> If a ball is thrown into the air with a velocity of 40 ft/s, its height (in feet) after t seconds is given by y = 40t - 16t2. Find the velocity when t = 2.
> Shown are graphs of the position functions of two runners, A and B, who run a 100-meter race and finish in a tie. a. Describe and compare how the runners run the race. b. At what time is the distance between the runners the greatest? c. At what time
> a. A particle starts by moving to the right along a horizontal line; the graph of its position function is shown in the figure. When is the particle moving to the right? Moving to the left? Standing still? b. Draw a graph of the velocity function.
> a. Find the slope of the tangent to the curve y = 1/√x at the point where x = a. b. Find equations of the tangent lines at the points (1, 1) and (4, 1/2). c. Graph the curve and both tangents on a common screen.
> Determine the infinite limit. 2 - x lim X- i (x – 1)?
> a. Find the slope of the tangent to the curve y = 3 + 4x2 - 2x3 at the point where x = a. b. Find equations of the tangent lines at the points (1, 5) and (2, 3). c. Graph the curve and both tangents on a common screen.
> Find an equation of the tangent line to the curve at the given point. y = 2x + 1/x + 2, (1, 1)
> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) = -o, lim f(x) = 5, lim f(x) = -5 X -00 X00
> For the function g whose graph is given, state the following. (a) lim g(x) (b) lim_g(x) X-00 (c) lim g(x) (d) lim g(x) x→2- (e) lim g(x) (f) The equations of the asymptotes x→2+ YA -1
> For the function f whose graph is given, state the following. e. The equations of the asymptotes (a) lim f(x) (b) lim f(x) X -0 (c) lim f(x) (d) lim f(x) yA 1 1
> a. Can the graph of y = f(x) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs. b. How many horizontal asymptotes can the graph of y = f(x) have? Sketch graphs to illustrate the possibilities.
> Explain in your own words the meaning of each of the following. (a) lim f(x) = 5 (b) lim f(x) = 3 X00 X -00
> a. Prove that And if these limits exist. b. Use part (a) and Exercise 65 to find lim f(x) = lim f(1/t) lim f(x) lim f(1/t) 1 lim x sin x→0+
> Formulate a precise definition of Then use your definition to prove that lim f(x) = X -00 lim (1 + x³) X -0 8.
> Use Definition 9 to prove that lim e* = 0.
> Determine the infinite limit. x + 1 lim x-5 x - 5
> Prove, using Definition 9, that lim x' = 0.
> Use Definition 8 to prove that 1 0. lim X-0 X
> a. How large do we have to take x so that 1/x2 b. Taking r = 2 in Theorem 5, we have the statement Prove this directly using Definition 7. 1 lim .2
> For the limit illustrate Definition 9 by finding a value of N that corresponds to M = 100. lim vx In x = 00 X00
> For the limit illustrate Definition 8 by finding values of N that correspond to ε = 0.1 and ε = 0.05. 1 — 3x lim = 3 -2 x² + 1 x- X -00
> For the limit illustrate Definition 7 by finding values of N that correspond to ε = 0.1 and ε = 0.05. 1 – 3x lim Vx2 + 1 -3
> Use a graph to find a number N such that 3x? + 1 if x> N then 1.5 < 0.05 | 2x2 + x + 1
> a. By graphing y = e-x/10 and y = 0.1 on a common screen, discover how large you need to make x so that e-x/10 < 0.1. b. Can you solve part (a) without using a graphing device?
> In Chapter 9 we will be able to show, under certain assumptions, that the velocity v(t) of a falling raindrop at time t is v(t) = v*(1 - e-gt/v*) Where g is the acceleration due to gravity and v* is the terminal velocity of the raindrop. a. Find limt →
> Determine the infinite limit. x + 1 lim X-5+ x - 5
> a. A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt after t minutes (in grams per liter) is C(t) = 30t/200 + t b. What happens to
> Find limx → ∞ f(x) if, for all x > 1, 10e* – 21 ) < I – x^ /x – 1 <f(x) 2e*
> By the end behavior of a function we mean the behavior of its values as x → ∞ and as x → -∞. a. Describe and compare the end behavior of the functions P(x) = 3x5 - 5x3 + 2x Q(x) = 3x5 by graphing both functions in the viewing rectangles [-2,
> a. Use the Squeeze Theorem to evaluate b. Graph f(x) = (sin x)/x. How many times does the graph cross the asymptote? sin x lim
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x+(x2 – 1)-(x + 2)
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = (3 - x)(1 + x)2(1 - x)4
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x3(x + 2)2(x - 1)
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x4 - x6
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = 2x3 - x4
> A function f is a ratio of quadratic functions and has a vertical asymptote x = 4 and just one x-intercept, x = 1. It is known that f has a removable discontinuity at x = -1 and limx →-1 f(x) = 2. Evaluate a. f(0) (b) lim f(x)
> a. Estimate the value of by graphing the function f(x) = (sin πx)/(sin πx). State your answer correct to two decimal places. b. Check your answer in part (a) by evaluating f(x) for values of x that approach 0. sin x lim X0 si
> The point Ps2, 21d lies on the curve y = 1/(1 – x). a. If Q is the point (x, 1/(1 - x)), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x: i. 1.5 ii. 1.9 iii. 1.99 iv. 1.999 v
> Find a formula for a function that has vertical asymptotes x = 1 and x = 3 and horizontal asymptote y = 1.
> Find a formula for a function f that satisfies the following conditions: lim f(x) = 0, lim f(x) = -, f(2) = 0, %3D lim f(x) = ∞, lim f(x) = -00
> Make a rough sketch of the curve y = xn (n an integer) for the following five cases: i. n = 0 ii. n > 0, n odd iii. n > 0, n even iv. n v. n Then use these sketches to find the following limits. (а) lim x^ X0+ (b) lim x" X0- (c) lim x" (d)
> Let P and Q be polynomials. Find if the degree of P is (a) less than the degree of Q and (b) greater than the degree of Q. P(x) lim Q(x)
> a. Graph the function f(x) = 2x2 + 1 /3x - 5 How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits b. By calculating values of f(x), give numerical estimates of the limits in part (a). c. Cal
> Estimate the horizontal asymptote of the function f(x) = 3x3 + 500x2/x3 + 500x2 + 100x + 2000 by graphing f for -10 ≤ x ≤ 10. Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?
> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = 2ex/ ex - 5
> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = x3 – x/x2 - 6x + 5
> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = 1 + x4/x2 - x4
> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = 2x2 + x – 1/x2 + x - 2
> a. By graphing the function f(x) = (cos 2x - cos x)/x2 and zooming in toward the point where the graph crosses the y-axis, estimate the value of limx→0 f(x). b. Check your answer in part (a) by evaluating f(x) for values of x that approach 0.
> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = 2x2 + 1/3x2 + 2x - 1
> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = 5 + 4x/x + 3
